To express the fractions 12/316 and 7/8 with a least common denominator (LCD), follow these steps:
Step 1: Determine the Denominators
The denominators of our fractions are 316 and 8.
Step 2: Find the Prime Factorization
Next, we need to find the prime factorization of each denominator:
- 316: It can be factored as follows:
- 316 = 2 × 158
- 158 = 2 × 79
- So, 316 = 2² × 79
- 8: The prime factorization is:
- 8 = 2 × 2 × 2 = 2³
Step 3: Identify the Least Common Denominator
The LCD is determined by taking the highest power of each prime factor found in the factorizations. Here, we find:
- For the prime factor 2: highest power is 2² (from 316) and 2³ (from 8), so we take 2³.
- For the prime factor 79: highest power is 79¹ (from 316).
Therefore, the LCD = 2³ × 79¹ = 8 × 79 = 632.
Step 4: Convert Each Fraction
Now we can convert each fraction to have the least common denominator of 632.
- For 12/316:
- To convert 12/316, we multiply both the numerator and the denominator by 2:
- 12 × 2 = 24
- 316 × 2 = 632
- So, 12/316 = 24/632.
- For 7/8:
- To convert 7/8, we multiply both the numerator and the denominator by 79:
- 7 × 79 = 553
- 8 × 79 = 632
- So, 7/8 = 553/632.
Final Result
Thus, both fractions expressed with the least common denominator are:
- 12/316 = 24/632
- 7/8 = 553/632
This way, we have successfully expressed both fractions using the least common denominator!